Zeeman's comparison theorem

In homological algebra, Zeeman's comparison theorem, introduced by Christopher Zeeman,[1] gives conditions for a morphism of spectral sequences to be an isomorphism.

Statement

Comparison theorem  Let be first quadrant spectral sequences of flat modules over a commutative ring and a morphism between them. Then any two of the following statements implies the third:

  1. is an isomorphism for every p.
  2. is an isomorphism for every q.
  3. is an isomorphism for every p, q.

Illustrative example

As an illustration, we sketch the proof of Borel's theorem, which says the cohomology ring of a classifying space is a polynomial ring.[2]

First of all, with G as a Lie group and with as coefficient ring, we have the Serre spectral sequence for the fibration . We have: since EG is contractible. We also have a theorem of Hopf stating that , an exterior algebra generated by finitely many homogeneous elements.

Next, we let be the spectral sequence whose second page is and whose nontrivial differentials on the r-th page are given by and the graded Leibniz rule. Let . Since the cohomology commutes with tensor products as we are working over a field, is again a spectral sequence such that . Then we let

Note, by definition, f gives the isomorphism A crucial point is that f is a "ring homomorphism"; this rests on the technical conditions that are "transgressive" (cf. Hatcher for detailed discussion on this matter.) After this technical point is taken care, we conclude: as ring by the comparison theorem; that is,

References

  1. Zeeman (1957).
  2. Hatcher, Theorem 1.34.
  • McCleary, John (2001), A User's Guide to Spectral Sequences, Cambridge Studies in Advanced Mathematics, vol. 58 (2nd ed.), Cambridge University Press, ISBN 978-0-521-56759-6, MR 1793722
  • Roitberg, Joseph; Hilton, Peter (1976), "On the Zeeman comparison theorem for the homology of quasi-nilpotent fibrations" (PDF), The Quarterly Journal of Mathematics, Second Series, 27 (108): 433–444, doi:10.1093/qmath/27.4.433, ISSN 0033-5606, MR 0431151
  • Zeeman, Erik Christopher (1957), "A proof of the comparison theorem for spectral sequences", Proc. Cambridge Philos. Soc., 53: 57–62, doi:10.1017/S0305004100031984, MR 0084769


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